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2
votes
1answer
96 views

Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model): \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} ...
0
votes
1answer
117 views

A kind of Discrete Fourier Transform

Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]_\mathbb{N}$ by $$DFT_z(k)=\frac{1}{N} \sum_{j=0}^{N-1} z_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th ...
3
votes
3answers
190 views

Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?

My question is as follows: Given ${{\lambda }_{1}},\,{{\lambda }_{2}},...,{{\lambda }_{n}}\in \mathbb{R}$ where $\underset{1\le j\le n-1}{\mathop{\min }}\,\left| {{\lambda }_{j+1}}-{{\lambda }_{j}} ...
1
vote
0answers
43 views

request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79 On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Hardy, G.H. I am not ...
0
votes
0answers
95 views

A question on a Fourier transform

(This may sound too simple a question to be asked in MO. But it is a research problem but not an exercise, so I post it here. If it is no appropriate, I will move it to Stackexchange.) Let $0 < a ...
5
votes
1answer
151 views

Largest area of a compactly supported positive definite function

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest area" $\int f\,dx$ that can be achieved? To be ...
-3
votes
1answer
390 views

A question about pointwise convergence of Fourier transform in $N$-dimensions

I am retreating back on this statement, after some explorations and calculation Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
6
votes
2answers
158 views

Reverse Hausdorff Young for nonnegative functions

The classical Hausdorff-Young inequality states that $$ \Vert \widehat{f} \Vert_{p'} \leq \Vert f \Vert_p \text{ for } 1 \leq p \leq 2. $$ For $p=2$, we even have equality due to Plancherel. If we ...
1
vote
0answers
54 views

Inversion of Fourier transform of a multivariate gamma distribution in polar form?

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
2
votes
0answers
77 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform ...
1
vote
1answer
124 views

Fourier approximation error in L^2 for piecewise continuous functions

Let $u:[0,2\pi)\to \mathbb{R}$ be the step function $$u(x) = \begin{cases} 1 & \text{if } x \in [0,\pi), \\ 0 & \text{if } x \in [\pi,2\pi) \end{cases}$$ By a direct computation, one ...
18
votes
1answer
394 views

Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant

Let $N$ be a positive integer; for simplicity I'm happy to assume it's an odd prime but I'm interested in the general case too. Let $f \colon \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let ...
5
votes
2answers
200 views

Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by $$ (\mathcal F u)(\xi)=\int e^{-2iπ ...
3
votes
1answer
150 views

A example on Fourier tranform of a continous compactly supported function

I am trying to find a continuous compactly supported function $f$ such that the Fourier transform $f^{ft}$ and derivative $(f^{ft})'$ of the $f^{ft}$ decay faster than exponential rates, that is ...
4
votes
2answers
110 views

Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$

Let $\alpha$ be an arbitrary real number and define \begin{align} \widehat{f}(\omega)=\left\{\begin{array}{ll} \omega^{-1+{\rm i}\alpha}, & \omega>1,\\ 0, & \textrm{otherwise}. \end{array} ...
3
votes
2answers
190 views

How do functions operate in a Sobolev space $H^{s}$?

Let $s>\frac{1}{2};$ and define a Sobolev space as follows: $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ Fact: Let $m$ ...
1
vote
1answer
92 views

Does the Fourier transform of a non-strictly positive real kernel $\Phi(t)$ always generate entire function with complex zeros? [closed]

Question (1) Does the Fourier transform of a non-strictly positive real kernel $f(t)$ always generate an entire function $g(z)$ with complex zeros? $$g(z)=\int_{-\infty}^{\infty}f(t) ...
1
vote
0answers
33 views

Cross-correlation of two functions which are not fixed

I am trying to cross-correlate two functions, but one of which is changing each 'step' of the cross-correlation. I want to cross-correlate T(f) and ...
0
votes
1answer
151 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it) It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...
0
votes
0answers
87 views

Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
0
votes
0answers
48 views

Godement-Jacquet and L-functions

Let $M_{r}(F)$ be the matrices with coefficients in a local nonarchimedean field $F$ and $q$ the cardinal of the residue field. We have a Fourier tansform on $M_{r}(F)$ with kernel ...
4
votes
0answers
57 views

Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of $e^x-1$. Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
3
votes
2answers
295 views

Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?

I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow. So I re-post it below. Riemann $\Xi(z)$ ...
5
votes
1answer
151 views

Are functions of moderate growth a bornological space?

I was thinking a bit about distribution theory the last weeks and stumbled across the following question: There are two natural locally convex topologies on the space of smooth functions of moderate ...
1
vote
0answers
48 views

Using FFT to aproximate a fuction [closed]

I am trying to use the FFT to approximate a given function. So i have 10 points xk that are given for example, if i use the FFT that will give me Xk. So now using the inverse FFT we can get the ...
8
votes
1answer
182 views

How is the propagator computed on an elliptic curve?

I've been struggling for a while now understanding why the propagator for the action $$ S(\varphi) = \int_E \partial \varphi \bar\partial\varphi + \frac{\lambda}{6}(\partial\varphi)^3 $$ on an ...
1
vote
0answers
52 views

Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting $$ (\exp_x^* ...
18
votes
2answers
2k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
1
vote
0answers
111 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
3
votes
0answers
116 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
2
votes
3answers
150 views

tranforms that lowers the number of variables of a function

Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
0
votes
1answer
171 views

Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
0
votes
1answer
124 views

Is $\int_0^\infty \sin(Kx)f_K(x)\,dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\,dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...
6
votes
2answers
216 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
2
votes
1answer
539 views

Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...
0
votes
0answers
20 views

function invariants under integral transforms

Some integral transforms have various invariants (Fourier transform, etc.), while others don't, such as Hilbert transform. I am wondering if there is any general method to find the invariant functions ...
3
votes
0answers
197 views

How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let: $\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\ G(x,y) &=& ...
2
votes
0answers
117 views

Positive Fourier coefficients for a function $f:\{+1,-1\}^n \to \mathbb R$

This is from my research in computer science where the Fourier transform over $GF(2)^n$ is a tool to study functions on the Boolean hypercube. For example, the majority function on 3 variables is ...
-3
votes
1answer
165 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
votes
0answers
111 views

Solvable PDEs and their Green's functions

I have a class of PDEs of the form $$ -\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0 $$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
1
vote
1answer
139 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
103 views

Relationship between Fourier series & DFT

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT. However, isn't there a flaw? When I integrate over the continuous ...
1
vote
1answer
264 views

When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
9
votes
0answers
274 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...
0
votes
1answer
128 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
3
votes
1answer
78 views

Generalized Radon transform (Relaxed sufficient condition for invertibility)

The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace. To be more ...
0
votes
0answers
57 views

Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution ) $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...
1
vote
0answers
119 views

How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
3
votes
1answer
126 views

Generalization of Lévy's continuity theorem for nuclear spaces

I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions ...
3
votes
0answers
97 views

Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...